Nourdienn Ouahab, John R. Graef, and Abdelghani Ouahab
Some Linear and Nonlinear Fractional Integral Inequalities
Dynamic Systems and Applications 33 (2024) 13-58
https://doi.org/10.46719/dsa2024.33.02
ABSTRACT.
In this paper the authors establish some fractional type generalizations of Gronwall and Bihari inequalities with singular kernels for ψ−Hilfer fractional derivatives. The linear and nonlinear version of Gronwall’s inequality for fractional integrals with respect to another function are also discussed. They prove some new Bihari type inequalities for ψ−Hilfer fractional integrals. Also, they give several versions of ψ−Hilfer-Gronwall inequalities with weak and double singularities. Delay Henry type inequalities are proved and using their characteristic method, they give some explicate estimates for these inequalities. In addition, they provide generalizations of a class of Henry-Gronwall type inequalities for variable fractional integrals that have been studied by Anastassion and Papanicolaou. For some of the proofs, an iterated method involving Holder’s and Young’s inequalities is used to simplify the more complex inequalities.
AMS (MOS) Subject Classification. 26A33, 26A42, 34A60, 54H25.
Key Words and Phrases. Fractional differential equations, Bihari inequality, fractional integral, ψ-Hilfer fractional derivative, variable fractional derivative, weak singularity, Henry type delay inequality.